Lift-and-Project Integrality Gaps for Santa Claus

This paper is devoted to the study of the MaxMinDegree Arborescence (MMDA) problem in layered directed graphs of depth $\ell\le O(\log n/\log \log n)$, which is a special case of the Santa Claus problem. Obtaining a poly-logarithmic approximation for MMDA in polynomial time is of high interest as it is the main obstacle towards the same guarantee for the general Santa Claus problem, which is itself a necessary condition to eventually improve the long-standing 2-approximation for makespan scheduling on unrelated machines by Lenstra, Shmoys, and Tardos [FOCS'87]. The only ways we have to solve the MMDA problem within an $O(\text{polylog}(n))$ factor is via a ``round-and-condition'' algorithm using the $(\ell-1)^{th}$ level of the Sherali-Adams hierarchy, or via a ``recursive greedy'' algorithm which also has quasi-polynomial time. However, very little is known about the limitations of these techniques, and it is even plausible that the round-and-condition algorithm could obtain the same approximation guarantee with only $1$ round of Sherali-Adams, which would imply a polynomial-time algorithm. As a main result, we construct an MMDA instance of depth $3$ for which an integrality gap of $n^{\Omega(1)}$ survives $1$ round of the Sherali-Adams hierarchy. This result is best possible since it is known that after only $2$ rounds the gap is at most poly-logarithmic on depth-3 graphs. Second, we show that our instance can be ``lifted'' via a simple trick to MMDA instances of any depth $\ell\in \Omega(1)\cap o(\log n/\log \log n)$, for which we conjecture that an integrality gap of $n^{\Omega(1/\ell)}$ survives $\Omega(\ell)$ rounds of Sherali-Adams. We show a number of intermediate results towards this conjecture, which also suggest that our construction is a significant challenge to the techniques used so far for Santa Claus.